The gluon condensate, $langle frac{alpha_s}{pi} G^2 rangle$, i.e. the leading order power correction in the operator product expansion of current correlators in QCD at short distances, is determined from $e^+ e^-$ annihilation data in the charm-quark
region. This determination is based on finite energy QCD sum rules, weighted by a suitable integration kernel to (i) account for potential quark-hadron duality violations, (ii) enhance the contribution of the well known first two narrow resonances, the $J/psi$ and the $psi(2S)$, while quenching substantially the data region beyond, and (iii) reinforce the role of the gluon condensate in the sum rules. By using a kernel exhibiting a singularity at the origin, the gluon condensate enters the Cauchy residue at the pole through the low energy QCD expansion of the vector current correlator. These features allow for a reasonably precise determination of the condensate, i.e. $langle frac{alpha_s}{pi} G^2 rangle =0.037 ,pm, 0.015 ;{mbox{GeV}}^4$.
Finite energy QCD sum rules with Legendre polynomial integration kernels are used to determine the heavy meson decay constant $f_{B_c}$, and revisit $f_B$ and $f_{B_s}$. Results exhibit excellent stability in a wide range of values of the integration
radius in the complex squared energy plane, and of the order of the Legendre polynomial. Results are $f_{B_c} = 528 pm 19$ MeV, $f_B = 186 pm 14$ MeV, and $f_{B_s} = 222 pm 12$ MeV.
Next to leading order corrections to the $SU(3) times SU(3)$ Gell-Mann-Oakes-Renner relation (GMOR) are obtained using weighted QCD Finite Energy Sum Rules (FESR) involving the pseudoscalar current correlator. Two types of integration kernels in the
FESR are used to suppress the contribution of the kaon radial excitations to the hadronic spectral function, one with local and the other with global constraints. The result for the pseudoscalar current correlator at zero momentum is $psi_5(0) = (2.8 pm 0.3) times 10^{-3} GeV^{4}$, leading to the chiral corrections to GMOR: $delta_K = (55 pm 5)%$. The resulting uncertainties are mostly due to variations in the upper limit of integration in the FESR, within the stability regions, and to a much lesser extent due to the uncertainties in the strong coupling and the strange quark mass. Higher order quark mass corrections, vacuum condensates, and the hadronic resonance sector play a negligible role in this determination. These results confirm an independent determination from chiral perturbation theory giving also very large corrections, i.e. roughly an order of magnitude larger than the corresponding corrections in chiral $SU(2) times SU(2)$. Combining these results with our previous determination of the corrections to GMOR in chiral $SU(2) times SU(2)$, $delta_pi$, we are able to determine two low energy constants of chiral perturbation theory, i.e. $L^r_8 = (1.0 pm 0.3) times 10^{-3}$, and $H^r_2 = - (4.7 pm 0.6) times 10^{-3}$, both at the scale of the $rho$-meson mass.
Experimental data on the total cross section of $e^+ e^-$ annihilation into hadrons are confronted with QCD and the operator product expansion using finite energy sum rules. Specifically, the power corrections in the operator product expansion, i.e.
the vacuum condensates, of dimension $d = 2$, 4 and 6 are determined using recent isospin $I=0+1$ data sets. Reasonably stable results are obtained which are compatible within errors with values from $tau$-decay. However, the rather large data uncertainties, together with the current value of the strong coupling constant, lead to very large errors in the condensates. It also appears that the separation into isovector and isoscalar pieces introduces additional uncertainties and errors. In contrast, the high precision $tau$-decay data of the ALEPH collaboration in the vector channel allows for a more precise determination of the condensates. This is in spite of QCD asymptotics not quite been reached at the end of the $tau$ spectrum. We point out that isospin violation is negligible in the integrated cross sections, unlike the case of individual channels.
The next to leading order chiral corrections to the $SU(2)times SU(2)$ Gell-Mann-Oakes-Renner (GMOR) relation are obtained using the pseudoscalar correlator to five-loop order in perturbative QCD, together with new finite energy sum rules (FESR) inco
rporating polynomial, Legendre type, integration kernels. The purpose of these kernels is to suppress hadronic contributions in the region where they are least known. This reduces considerably the systematic uncertainties arising from the lack of direct experimental information on the hadronic resonance spectral function. Three different methods are used to compute the FESR contour integral in the complex energy (squared) s-plane, i.e. Fixed Order Perturbation Theory, Contour Improved Perturbation Theory, and a fixed renormalization scale scheme. We obtain for the corrections to the GMOR relation, $delta_pi$, the value $delta_pi = (6.2, pm 1.6)%$. This result is substantially more accurate than previous determinations based on QCD sum rules; it is also more reliable as it is basically free of systematic uncertainties. It implies a light quark condensate $<0|bar{u} u|0> simeq <0|bar{d} d|0> equiv <0|bar{q} q|0>|_{2,mathrm{GeV}} = (- 267 pm 5 MeV)^3$. As a byproduct, the chiral perturbation theory (unphysical) low energy constant $H^r_2$ is predicted to be $H^r_2 ( u_chi = M_rho) = - (5.1 pm 1.8)times 10^{-3}$, or $H^r_2 ( u_chi = M_eta) = - (5.7 pm 2.0)times 10^{-3}$.
Thermal Hilbert moment QCD sum rules are used to obtain the temperature dependence of the hadronic parameters of charmonium in the vector channel, i.e. the $J$ / $psi$ resonance mass, coupling (leptonic decay constant), total width, and continuum thr
eshold. The continuum threshold $s_0$, which signals the end of the resonance region and the onset of perturbative QCD (PQCD), behaves as in all other hadronic channels, i.e. it decreases with increasing temperature until it reaches the PQCD threshold $s_0 = 4 m_Q^2$, with $m_Q$ the charm quark mass, at $Tsimeq 1.22 T_c$. The rest of the hadronic parameters behave very differently from those of light-light and heavy-light quark systems. The $J$ / $psi$ mass is essentially constant in a wide range of temperatures, while the total width grows with temperature up to $T simeq 1.04 T_c$ beyond which it decreases sharply with increasing T. The resonance coupling is also initially constant and then begins to increase monotonically around $T simeq T_c$. This behaviour of the total width and of the leptonic decay constant provides a strong indication that the $J$ / $psi$ resonance might survive beyond the critical temperature for deconfinement.
Quantum fluctuations in the QED vacuum generate non-linear effects, such as peculiar induced electromagnetic fields. In particular, we show here that an electrically neutral particle, possessing a magnetic dipole moment, develops an induced electric
dipole-type moment with unusual angular dependence, when immersed in a quasistatic, constant external electric field. The calculation of this effect is done in the framework of the Euler-Heisenberg effective QED Lagrangian, corresponding to the weak field asymptotic expansion of the effective action to one-loop order. It is argued that the neutron might be a good candidate to probe this signal of non-linearity in QED.
The up and down quark masses are determined from an optimized QCD Finite Energy Sum Rule (FESR) involving the correlator of axial-vector divergences, to five loop order in Perturbative QCD (PQCD), and including leading non-perturbative QCD and higher
order quark mass corrections. This FESR is designed to reduce considerably the systematic uncertainties arising from the (unmeasured) hadronic resonance sector, which in this framework contributes less than 3-4% to the quark mass. This is achieved by introducing an integration kernel in the form of a second degree polynomial, restricted to vanish at the peak of the two lowest lying resonances. The driving hadronic contribution is then the pion pole, with parameters well known from experiment. The determination is done in the framework of Contour Improved Perturbation Theory (CIPT), which exhibits a very good convergence, leading to a remarkably stable result in the unusually wide window $s_0 = 1.0 - 4.0 {GeV}^2$, where $s_0$ is the radius of the integration contour in the complex energy (squared) plane. The results are: $m_u(Q= 2 {GeV}) = 2.9 pm 0.2 $ MeV, $m_d(Q= 2 {GeV}) = 5.3 pm 0.4$ MeV, and $(m_u + m_d)/2 = 4.1 pm 0.2$ Mev (at a scale Q=2 GeV).
The Euler-Heisenberg effective Lagrangian is used to obtain general expressions for electric and magnetic fields induced by non-linearity, to leading order in the non-linear expansion parameter, and for quasistatic situations. These expressions are t
hen used to compute the induced electromagnetic fields due to a spherical shell with uniform charge distribution on the surface, in the presence of an external constant magnetic field. The induced electric field contains several multipole terms with unusual angular dependences. Most importantly, the leading term of the induced magnetic field is due to an induced magnetic dipole moment.
The light quark masses are determined using a new QCD Finite Energy Sum Rule (FESR) in the pseudoscalar channel. This FESR involves an integration kernel designed to reduce considerably the contribution of the (unmeasured) hadronic resonance spectral
functions. The QCD sector of the FESR includes perturbative QCD (PQCD) to five loop order, and the leading non-perturbative terms. In the hadronic sector the dominant contribution is from the pseudoscalar meson pole. Using Contour Improved Perturbation Theory (CIPT) the results for the quark masses at a scale of 2 GeV are $m_u(Q= 2 {GeV}) = 2.9 pm 0.2 {MeV}$, $m_d(Q= 2 {GeV}) = 5.3 pm 0.4 {MeV}$, and $m_s(Q= 2 {GeV}) = 102 pm 8 {MeV}$, for $Lambda = 381 pm 16 {MeV}$, corresponding to $alpha_s(M_tau^2) = 0.344 pm0.009$. In this framework the systematic uncertainty in the quark masses from the unmeasured hadronic resonance spectral function amounts to less than 2 - 3 %. The remaining uncertainties above arise from those in $Lambda$, the unknown six-loop PQCD contribution, and the gluon condensate, which are all potentially subject to improvement.
